State-Level High School Completion Rates : Concepts , Measures , and Trends * Version : July 2004

Since the mid 1970s the national rate at which incoming 9th graders have completed high school has fallen slowly but steadily; this is also true in 41 states. In 2002, about three in every four students who might have completed high school actually did so; in some states this figure is substantially lower. In this paper I review state-level measures of high school completion rates and describe and validate a new measure that reports these rates for 1975 through 2002. Existing measures based on the Current Population Survey are conceptually imperfect and statistically unreliable. Measures based on Common Core Data (CCD) dropout information are unavailable for many states and have different conceptual weaknesses. Existing measures based on CCD enrollment and completion data are systematically biased by migration, changes in cohort size, and/or grade retention. The new CCD-based measure described here is considerably less biased, performs differently in empirical analyses, and gives a different picture of the dropout situation across states and over time.


State-Level High School Completion Rates: Concepts, Measures, and Trends
Each fall, and in every state, a new cohort of students enters high school for the first time.A few years later a portion of each cohort successfully completes high school and the rest does not.At first glance, the task of quantifying the proportion of entering students in each state who go on to complete high school seems straightforward.Years of effort by academic and government researchers has proven otherwise.
There are at least three compelling reasons to develop, analyze, and disseminate state-level high school completion rates.The first is that high school completion is extremely important both socially and economically for students and for the states in which they reside.Consequently, it is inherently worth asking how successful students are in each state at reaching this critical educational milestone.Second, as part of the provisions of the 2002 No Child Left Behind legislation states must meet annual yearly progress (AYP) goals.For secondary education, states' definitions of AYP are mandated to include "graduation rates for public secondary school students (defined as the percentage of students who graduate from secondary school with a regular diploma in the standard number of years)" [Sec 1111(b)(2)(D)(i)].Third, researchers who are interested in the impact of state education policy initiatives-such as the implementation of mandatory high school exit examinations or changes in course requirements for high school graduation-need reliable and valid state-level high school completion rates in order to come to sound conclusions.
In this paper I review and critique existing measures of state-level high school completion rates and describe a new measure that reports state-level high school completion rates for 1973 through 2000.This new measure is more conceptually sound and less empirically biased than existing measures, performs differently in empirical analyses, and yields a different picture of differences across states and over time in state-level high school completion rates.

CONCEPTUAL AND TECHNICAL GOALS
My goal is to develop a state-level measure of the rate at which incoming 9 th grade students complete public high school; I do not count holders of General Educational Development (GED) certificates as high school completers.This conceptualization ignores high school dropout/completion that occurs before or long after the high school years and it also ignores private high school completers. 1 The state-level high school completion measure that I create is thus not a measure of the proportion of people who eventually earn any secondary education credential; it is a measure of the rate at which people succeed in completing public high school. 2llowing Hauser (1997: 159), there are several desirable technical properties of any good measure of the rate of high school completion.Three are particularly relevant here.First, such measures should have face validity.For example, if every student in a particular incoming cohort in a particular state goes on to complete high school then the high school completion rate for that cohort in that state should equal 100%.As I will demonstrate, widely-used measures of state-level high school completion rates fail to meet this basic standard.Second, such measures should "be consistent with a reasonable understanding of the process or processes that it purports to measure" and "should pertain to a well-defined population and set of events."For present purposes, a good measure of state-level high school completion rates should pertain to specific cohorts of incoming students 1 Below I discuss the implications of ignoring private high school completers.Briefly, patterns of public high school completion over time and across states as portrayed by the measure developed in this paper would not be meaningfully different were I to include private school completers. 2 The measure that I create is not a four-year high school completion rate measure.It is a measure of the rate at which incoming 9 th grade students complete public high school, regardless of how long it takes them to do so.This means that my measure does not squarely meet the AYP definition described above, which requires a measure of four year completion rates.
(e.g., students who first entered the 9 th grade in 1988) and should adequately account for such issues as migration, changes over time in the size of incoming cohorts, mortality, and grade retention.Finally, such measures should be statistically reliable: Good measures of state-level high school completion rates should be based on enough observations to allow statistically sound comparisons across states and across cohorts of the rate at which incoming students complete high school.

CURRENT MEASURES
Existing measures of annual state-level high school completion and dropout rates come from one of only two sources of data: the Current Population Surveys (CPS) and the Common Core of Data (CCD). 3The CPS is a monthly survey of more than 50,000 households, and is conducted by the Bureau of the Census for the Bureau of Labor Statistics.Households are selected in such a way that it is possible to make generalizations about the nation as a whole, and in recent years about individual states and other specific geographic areas.Individuals in the CPS are broadly representative of the civilian, non-institutionalized population of the United States.In addition to the basic demographic and labor force questions that are included in each monthly CPS survey, questions on selected topics are included in most months.Since 1968 the October CPS has obtained basic monthly data as well as information about school enrollment-including current enrollment status, public versus private school enrollment, grade attending if enrolled, most recent year of enrollment, enrollment status in the preceding October, grade of enrollment in the preceding October, and high school completion status.In recent years the October CPS has also ascertained whether high school completers earned diplomas or GED certificates. 3State-level high school completion and dropout rates can also be computed from decennial census data-but only for every tenth year-and shortly from the American Community Survey.I am referring to data that allows annual state-level estimates.

Measures Based On CPS Data
Published national estimates of high school completion and dropout have historically been based on CPS data.CPS-derived event dropout rates report the percentage of students in a given age range who leave school each year without first obtaining a diploma or GED.For example, 4.8% of 15 to 24 year olds who were enrolled in high school in October 1999 left school by October of 2000 without obtaining a diploma or GED.CPS-derived status dropout rates report the percentage of people within an age range-typically ages 16 to 24-who are not enrolled in school and who have not obtained a diploma or GED.In October 2000, about 10.9% of 16 to 24 year olds were not enrolled in school and did not have a diploma or GED (U.S.Department of Education 2001a).
For present purposes there are a number of conceptual and technical problems with CPSderived measures of high school dropout and completion, particularly when computed at the state level.First and foremost, the sample sizes for some states are not large enough to produce reliable estimates of rates of high school completion or dropout (Kaufman 2001; U.S. Department of Education 2000).Even when data are aggregated across years-for example, in the Annie E. Casey Foundation's Kids Count (2004) measure-the standard errors of estimates for some states are frequently so large that it is difficult to make comparisons across states or over time.What is more, by aggregating across years the resulting measure no longer pertains to specific cohorts of incoming students; this is a serious problem for researchers interested in the effects of state education policy reforms that typically take effect for specific cohorts of students.
Second, until 1987 it was not possible to distinguish high school completers from GED recipients; since 1988 October CPS respondents who recently completed high school have been asked whether they obtained a diploma or GED, but there are concerns about the quality of the resulting data (Chaplin 2002; Kaufman 2001).Third, as noted by Greene (2002: 7), "[status] dropout statistics derived from the Current Population Survey are based on young people who live in an area but who may not have gone to high school in that area."To the extent that young people move from state to state, CPS-based state-level high school dropout rates-particularly status dropout rates based on 16 to 24 year olds-may be of questionable validity. 4Fourth, some observers have expressed concern about coverage bias in the CPS, particularly for race/ethnic minorities.The CPS is representative of the civilian, non-institutionalized population of the United States, and so young people who are incarcerated or in the military are not represented.To the extent that these populations differ from the rest of the population with respect to frequency and method of high school completion, there is the potential for bias in estimates.Finally, substantial changes over time in CPS questionnaire design, administration, and survey items have made year-to-year comparisons difficult (Hauser 1997; Kaufman 2001).For these reasons, the state-level high school completion rate measure that I construct is based primarily on CCD data, not on CPS data.However, as described below I use supplementary information from the CPS to overcome some of the limitations of CCD-based measures.
In the sections that follow I describe existing techniques for estimating state-level high school completion rates using CCD data.Each technique has serious conceptual shortcomings for my purposes, and below I demonstrate that each technique also yields systematically biased estimates.

Measures Based on Common Core Data I: NCES Completion Rate (NCES)
Since the early 1990s NCES has asked state education agencies to report the number of students who drop out in each year; state-level dropout rates have been part of the CCD beginning with the 1992-1993 data collection (U.S.Department of Education 2002b) which asked about the 1991-1992 academic year.On October 1 of each year the NCES asks states to define as a dropout any student who (1) was enrolled at any point during the previous academic year, (2) was not enrolled at the beginning of the current academic year, and (3) has not graduated or completed an approved education program (e.g., obtained a GED).Students are not counted as dropouts if they died, are absent from school for reasons of health or temporary suspension, or if they transfer to another jurisdiction.
NCES then computes annual event dropout rates by dividing the number of 9 th through 12 th grade dropouts by the total 9 th through 12 th grade enrollment as of October 1.Using these dropout data, NCES also reports a 4-year high school completion rate as: (2) Indeed Haney (2000; 2001) has used exactly such a measure in highly publicized and much-cited recent work on the impact of high school exit examinations on rates of high school completion.The BCR is purportedly a measure of the overall high school completion rate, not a measure of the fouryear high school completion rate.However, the BCR has at least four problems, each of which induces systematic bias in estimated state-level completion rates.
The first problem with the BCR has to do with migration.Students who appear as 9 th graders in a state in the fall of academic year X may move to another state before the spring of academic year X+3; they may be replaced by (a smaller or larger number of) students who are counted among the number of high school completers in the spring of academic year X+3 but who lived in another state in the fall of academic year X.A second problem with the BCR has to do with grade retention.If we are interested in the number of incoming 9 th graders who go on to complete high school, then measures like the BCR are problematic to the extent that the denominator includes 9 th graders who are enrolled in the 9 th grade in more than one academic year; essentially, such measures count retained 9 th graders in the denominator for more than one year but in the numerator a maximum of one time.As I will demonstrate below, each of these first two issues call into question the validity of the BCR as a measure of high school completion rates.In recent work, Haney and colleagues (2004)   have tried to overcome the grade retention problem by using the number of 8 th graders enrolled in academic year X-4 as the denominator for the BCR.Since fewer students are made to repeat 8 th grade than are made to repeat 9 th grade, this partially alleviates the grade retention bias; however, the longer time horizon exacerbates the migration bias.A third problem with the BCR has to do with mortality: Students who die before they complete high school are counted as dropouts.A fourth problem has to do with students who are in un-graded (frequently special education) programs and who might be counted as high school completers in the numerator but not as 9th graders in the denominator.Because less than 0.2% of young people die during the modal ages of high school enrollment (Arias 2002)   (5) "Smoothing" the 9 th grade enrollments is designed to minimize the bias introduced by grade retention, while the migration adjustment is designed to account for bias introduced by net migration between academic years X-3 and X.As I will show below, these adjustments produce valid state-level completion rates only under very specific (and relatively unlikely) demographic circumstances.Although "Greene's Method" is an effort to adjust for the two major problems in completion rates like the BCR, as I show below the details of the ACR actually produce less valid results than the BCR under most circumstances.
What is more, because states differ among themselves and over time with respect to whether and how they differentiate between "regular diplomas," "other diplomas," and "other high school completers," Greene and colleagues introduce a serious new form of bias by restricting the numerator to "regular diplomas."For example, in the CCD data the number of regular diplomas issued in California rose from 259,071 in 1996 to 311,818 in 1997-apparently reflecting a dramatic one year change in the number of high school completers.However, the total number of high school completers in California rose from 304,038 in 1996 to only 311,818 in 1997-reflecting much less change.
This is because the CCD data report that 44,967 "other diplomas" were issued in California in 1996, while none were issued in California in 1997.It is clear that this is a change in classification, not a change in reality.In producing our own state-level completion rates we follow NCES and other researchers by combining these categories of completers (and by continuing to exclude GED recipients from the category of high school completers).

Measures Based on Common Core Data IV: Cumulative Promotion Index (CPI)
Swanson ( 2003) recently proposed a new method for calculating state-level four-year high school completion rates which "approximates the probability that a student entering the 9th grade will complete high school on time with a regular diploma.It does this by representing high school graduation rate [sic] as a stepwise process composed of three grade-to-grade promotion transitions ( 9to 10, 10 to 11, and 11to 12) in addition to the ultimate high school graduation event (grade 12 to diploma)" (Pg.14).Specifically, the Cumulative Promotion Index is: E equals the number of 12 th graders enrolled in the fall of academic year X.The author notes that this approach "estimates the likelihood of a 9 th grader from a particular district completing high school with a regular diploma in four years given the conditions in that district during the [given] school year" (emphasis theirs).Swanson (2003) argues that this measure has the virtues of being timely and reflective of current education system performance because it requires data from only two academic years.As I will demonstrate below, the CPI is systematically biased except when there is no net student migration between geographic units.What is more, the CPI shares with the ACR the technical weakness of including only regular diploma recipients in the numerator; in his defense, Swanson's (2003) includes only regular diploma recipients in his four-year high school completion rate because this is what is required under the AYP provisions of No Child Left Behind.

EVALUATING MEASURES BASED ON COMMON CORE DATA
Table 1 presents a series of simulations of enrollment counts, high school completer counts, and high school completion rates in one geographic area over ten academic years.I include the BCR with 8 th grade enrollments in the denominator, the BCR with 9 th grade enrollment in the denominator, the ACR, the CPI, and the new Estimated Completion Rate (ECR) that I describe below.For demonstration purposes, all simulations stipulate that every single student completes high school.By design, then, valid measures of overall high school completion rates should report a 100% completion rate for every academic year; four-year completion rate measures like the CPI may be less than 100% in the presence of grade retention (which would delay students' graduation).The simulations differ with respect to assumptions about (1) changes over time in the numbers of incoming 8 th graders, (2) net migration rates, and (3) grade retention rates.Each simulation begins with 1,000 students entering the 8 th grade for the first time in the fall of the 1994-1995 academic year and follows that and subsequent cohorts of students over ten academic years under a variety of assumptions about cohort sizes, net migration, and grade retention.
Panel A of Table 1 simulates a situation in which the size of the incoming 8 th grade cohort increases by 2% annually, from 1,000 in 1994-1995 to 1,020 in 1995-1996 and so forth; there is no net migration, no students are ever retained in grade, and (as always) all students complete high school.Given these parameters, all of the 1,000 students who enter 8 th grade in the fall of 1994 progress to the 9 th grade in the fall of 1995, to the 10 th grade in the fall of 1996, to the 11 th grade in the fall of 1997, and to the 12 th grade in the fall of 1998, and all 1,000 receive regular diplomas in the spring of 1999.The incoming cohort of 8 th graders in fall 1995 enjoys similar success, such that all 1,020 obtain regular diplomas in spring 2000.As reported at the bottom of the panel, each of the CCD-based completion rates correctly reports a 100% high school completion rate-except for Greene and Winter's (2002) ACR.The ACR equals 106% under these conditions.In general, if the annual proportional change in the size of 8 th grade cohorts equals X (e.g., 0.02 in Panel A), then the ACR equals the true rate times (1+X) 3 .1 simulates a situation in which the net migration rate equals +1% at each grade level, such that the number of students in each grade and in each year grows by 1% during the course of the academic year because more students move into the state (either from another state or from abroad) than leave it.Here there is no annual change in the size of incoming cohorts of 8 th graders, and no students are ever retained in grade.Under this scenario, each of the CCD-based high school completion rates described above is biased; the BCR with the number of 8 th graders in the denominator yields a 105% completion rate, while the other measures each yield a 104% completion rate.In general, if the annual net migration rate is expressed as proportion Y, then the ACR, the CPI, and the BCR with 9 th grade enrollments in the denominator yield completion rates that equal the true rate times (1+Y) 4 .Interestingly, despite the "migration adjustment" detailed in Equation 5Greene and Winter's (2002) ACR shares the same migration-related bias as the BCR.The simulation in Panel C is the same as the simulation in Panel B except that the net migration rate is now -1%.Here each of the CCD-based high school completion rates is downwardly biased.

Panel B of Table
The simulations in Panels A, B, and C of Table 1 make two general points about the role of migration and changes in cohort size in biasing these several state-level high school completion rate measures.First, whether these rates are biased upward or downward depends heavily on net migration rates.If more students move into a state than leave it between 8 th grade and 12 th grade, then these completion rates are overstated.If more students leave a state than move into it, then these completion rates are underestimated.Second, Greene and Winter's (2002) ACR-but not the BCR or CPI-is also biased by changes over time in the size of incoming cohorts of 8 th graders.
Panel D of Table 1 presents a simulation in which the percentage of 9 th graders made to repeat the 9 th grade begins at 3% and then rises over time.Here there is no annual change in the size of incoming cohorts of 8 th graders, there is no net migration, and (as always) every student completes high school.Although 1,000 students enter the 9 th grade for the first time in each academic year, not all of them move on to the 10 th grade in the succeeding academic year.Consequently, the observed number of 9 th graders in each year is higher than the number of new, incoming 9 th graders in that year.Except for the BCR that uses 8 th grade enrollments in the denominator, each of the CCD-based measures of overall high school complete rates described above is downwardly biased when any 9 th graders are retained-even though all incoming 9 th graders end up completing high school.5This is because each measure (with one exception) counts retained students in their denominator twice (once in the year in which they first entered the 9 th grade and once in the following year) but in their numerator only once.In general, the ACR and the BCR with 8 th grade enrollments in the denominator are downwardly biased when any students are made to repeat the 8 th grade.The ACR and the BCR with 9 th grade enrollments in the denominator are downwardly biased when any students are made to repeat the 9 th grade, and the ACR is downwardly biased when any students are made to repeat the 10 th grade.The fact that more students repeat 9 th grade than any other high school grade-combined with recent claims that rates of 9 th grade retention are increasing (Haney et al. 2004)-is troubling, since retention in the 9 th grade has the most deleterious consequences for the validity of the ACR and the BCR with 9 th grade enrollments in the denominator.
The simulations in Table 1 make the point that CCD-based high school completion rates like those reviewed above are biased except when there is no grade retention, when there is no net migration, and when cohort sizes remain stable.The direction and magnitude of systematic bias in the BCR, ACR, and CPI depend on the configuration of demographic and grade retention patterns in particular states in particular years.Beyond misrepresenting the absolute rates of high school completion, this means that these measures also misrepresent differences across states and trends over time in high school completion rates-unless net migration, the size of incoming cohorts of 8 th graders, and rates of grade retention (particularly in the 9 th grade) remain stable over time and across states.What is more, as I will show below these alternate measures produce substantively different results in empirical analyses.

A NEW METHOD FOR MEASURING STATES' HIGH SCHOOL COMPLETION RATES
In this section I describe a new CCD-based measure of high school complete rates-labeled Estimated Completion Rates (ECR)-that can presently be computed for the graduating classes of 1973 to 2000; revised completion counts for spring 2001 and beyond are not yet available in the CCD.As shown in Table 1, this new measure produces unbiased estimates of the rate of high school completion regardless of changes over time in incoming cohort sizes, migration patterns, or grade retention rates.After describing the construction of this new measure I employ it for the purposes of comparing high school completion rates across states and over time.
The ECR conceptually represents the proportion of incoming public school 9 th graders in a particular state and in a particular year who go on to complete high school (except via GED certification).The ECR is computed as The ECR is essentially the BCR with adjustments to the denominator to account for retention in the 9 th grade and for migration.The goal of these adjustments is to cause the denominator to represent the number of individuals in the cohort who are at risk of completing high school in the spring of academic year X.Ninth graders in state Z in the fall of academic year X-3 who are made to repeat the 9 th grade or who move to a different state are not at risk of completing high school in state Z in the spring of academic year X.Conversely, the population of students at risk of completing high school in state Z in the spring of academic year X includes in-migrants who were not counted among the 9 th graders in state Z in the fall of academic year X-3.
For reasons described above, the numerator in Equation 7is the total number of public high school completers (excluding GED recipients), regardless of whether completers earned regular diplomas, earned "other diplomas," or completed high school in some other way.Again, historically about 99% of completers have earned regular diplomas.The denominator begins with the number of public school 9 th graders enrolled in the fall of academic year X-3, but adjusts this figure in two ways to account for retention in the 9 th grade and for migration.
The adjustment for retention in the 9 th grade is based on the estimated percentage of 9 th graders in a particular state in the fall of a particular year that is in 9 th grade for the first time.These estimates are derived from 1968 through 1996 data from the October CPS.Using these data I select students who were enrolled in public schools in one of the 50 states or the District of Columbia. 6e 9 th grade retention adjustment to the denominator in Equation 7 is based on a comparison of the age distribution of 8 th graders in one October to the age distribution of 9 th graders in the following October.In the fall of an academic year, the vast majority of 8 th and 9 th graders are 13 and 14 years old, respectively.I begin by defining students as overage for grade if they are 14 or older in 8 th grade or 15 or older in 9 th grade.After computing the proportion of 8 th and 9 th graders who are overage for grade in each October, I assume that the growth in the proportion of students who are overage for grade is due to grade retention.For example, in California in 1968 I observe that 16.4% of 8 th graders were overage for grade.In 1969, however, 19.7% of California 9 th graders were overage for grade.I thus estimate that 19.7 -16.4 = 3.3% of California 9 th graders are repeating the 9 th grade, or that 96.7% of all 9 th graders in California in 1969 were in 9 th grade for the first time.This estimate is subject to random error, but as I describe below the degree of bias in this measure is likely small; even with some degree of random error, this estimate of grade retention is preferable to either ignoring grade retention altogether or to employing demonstrably biased estimates of grade retention.In order to minimize the degree of random error in these estimates of 9 th grade retention (which are based on CPS data which have small sample sizes in some states in some years) I have taken three additional steps in constructing the 9 th grade retention adjustment to the denominator in Equation 7. First, I have constrained grade retention rates to be 0 or greater; small sample sizes occasionally yielded negative estimates.Second, I have aggregated some smaller states into geographically proximate state groups. 7This step has the consequence of forcing 9 th grade retention rates to be equal across some states, but it has the advantage of yielding more reliable estimates.Third, I have used five year moving averages to smooth grade retention rates over time; again, the goal is to produce more reliable estimates.In practice, this means that the estimated 9 th grade retention rate for These estimates of 9 th grade retention are imperfect in at least two respects.First, they ignore migration.If the proportion of students who are overage for 8 th grade in state Z in academic year X differs from the proportion of students who are overage for 9 th grade in that state one year later, this difference may in part be attributable to selective migration.For this problem to seriously bias these estimates, however, there would need to be strong relationships between students' ages and their propensity to migrate, such that overage-for-grade students are much more (or less) likely to migrate than their younger classmates.The second potential problem with these estimates has to do with dropout.If students who are overage for 8 th grade drop out of school before being counted among the population of overage for 9 th grade students the following year, then estimated 9 th grade retention rates are downwardly biased.The October CPS data show that more than 98% of all 14 years olds and more than 98% of all 15 years olds have been enrolled in school in any particular year since 1968; these figures fall to about 95% for 16 year olds and about 88% for 17 year olds.Because the vast majority of overage for 8 th grade students are 14 and the vast majority of overage for 9 th grade students are 15, the magnitude of this bias in 9 th grade retention rates is very likely small.I have taken one additional step to investigate the validity of these measures of 9 th grade retention.Since 1988 the October CPS has included a measure of the grade in which students were enrolled in the previous October.Since the October CPS also includes a measure of the grade in which students are currently enrolled it is possible to use these data to estimate the proportion of enrolled, public school 9 th graders in each state and in each year (since 1988) who were attending the 9 th grade for the first time in that year.For example, according to the October CPS there were 484,372 students in 9 th grade in California in 2000.However, 9,299 of these students were also enrolled in 9 th grade in the preceding October.Rates of 9 th grade retention derived as such are similar to the rates described above, both in terms of the absolute proportions of students retained in 9 th grade and in states' relative grade retention rates.This gives some assurance that the basic estimation scheme for 9 th grade retention rates yields credible results.
The adjustment for migration in the denominator of Equation 7 is based on a comparison of the total population of 17 year olds in a state on July 1 of a particular year to the total population of The first issue is that these migration estimates pertain to the net change in the population size of all 14 year olds over the ensuing three years-not to net change in the population size of all 14 year olds 8 Although I refer to these as estimates of net migration, these figures actually represent the influence of both net migration and mortality; indeed only migration and mortality can lead to differences between the numbers of 14 year olds in a state in one year and the numbers of 17 year olds in that state 3 years later.students.However, as described above more than 98% of 14 year olds are enrolled in school; consequently, the empirical biases resulting from this conceptual issue are likely trivial.The second issue is that these estimates cover only three years of migration between ages 14 and 17.Surely there is some migration among high school students between ages 17 and 18, and this migration is missed in my estimates.Although it is possible to use the Census Bureau's population figures to estimate migration between ages 14 and 18, these estimates would capture a great deal of inter-state migration among 18 year olds who are moving for the purpose of attending college out of state.Consequently, my estimated migration rates are likely a bit conservative (although the direction of bias depends on whether net migration is positive or negative within states).The third issue is that this technique counts international in-migrants who come to the U.S. between ages 14 and 17-but never enroll in high school-as non-completers.As I discuss in detail below, this exerts modest downward bias on estimated completion rates, particularly in states with high levels of international in-migration.In the end, however, this small degree of bias is certainly preferable to the considerable bias introduced by ignoring migration altogether as most previous measures have done.
Above and beyond the technical issues involved in calculating the 9 th grade retention and migration adjustments, a potential technical weakness of the ECR more generally concerns its treatment of students who are made to repeat any high school grade other than grade 9. Students enrolled in the 9 th grade in academic year X-3 who are made to repeat one grade during high school are not at risk of completing high school in the spring of academic year X-but they may still complete high school in academic year X+1.Consequently, the ECR may seem like a downwardly biased estimator of high school completion rates.However, consider the fact that students enrolled in the 9 th grade in academic year X-3 who are made to repeat one grade during high school are at risk of completing high school in the spring of academic year X+1.What this means is that as long as grade retention rates remain stable-regardless of their absolute levels-the ECR is unbiased.What is more, be-cause of the 9 th grade retention adjustment the ECR is not biased by changes in 9 th grade retention rates ----only by changes in retention rates in grades 10 through 12. Simulations (not shown, but available upon request) demonstrate that extreme single-year changes in the rate at which 10 th through 12 th graders are retained produce very modest downward bias in the ECR.For example, if we assume that 5% of all 10 th , 11 th , and 12 th graders are retained in academic year X, and that the retention rate rises by 0.5% annually in each grade-such that 5.5% of all 10 th , 11 th , and 12 th graders are retained in academic year X+1, 6% of all 10 th , 11 th , and 12 th graders are retained in academic year X+2, and so forth-the ECR is downwardly biased by about 2%.In short, extreme annual changes in grade retention rates in grades 10 through 12-but not extreme grade retention rates themselvesproduce modest downward biases in the ECR.

THE ECR: AN EXAMPLE
To illustrate the computation of the ECR in practice, consider that there were 70,811 students in 9 th grade in Massachusetts in 1996 and that there were 52,950 high school completers in that state in 2000 (all according to CCD data).The BCR in this case would equal

= = =
However, I estimate that 5.3% of Massachusetts 9 th graders in the fall of 1996 were repeating that grade, such that only 70,811 x 0.947 = 67,058 were newly enrolled 9 th graders.Moreover, the population of 17 year olds in Massachusetts in 1999 was 2.3% larger than the population of 14 year olds in that state in 1996.Consequently, I estimate that 67,058 x 1.023 = 68,600 individuals were actually at risk of completing high school in Massachusetts in the spring of 2000.The ECR thus equals

VALIDATING THE ECR
Although the ECR is designed to produce valid estimates of state-level high school completion rates, it is worth asking how national estimates derived from the ECR compare to high school completion rates derived from longitudinal surveys of students-surveys in which we actually observe the percentage of students who complete high school among those at risk of doing so.For example, the National Educational Longitudinal Study of 1988 (NELS-88) is a longitudinal study of more than 25,000 students who were 8 th graders in the spring of 1988 (U.S.Department of Education 2002c).If we restrict the NELS-88 sample to public school students who were included in the 1994 follow-up survey, 9 we find that 75.4% of respondents completed high school (except via GED certification) in 1992 (which is to say, within four academic years). 10For the graduating class of 1992 the ECR equals 73.7%.However, because the migration component of the ECR-which equals +3.8% in 1992-reflects patterns of international migration that are not captured in NELS-88, 11 a more reasonable comparison would be to the ECR without including the migration component.For 1992, the ECR without including the migration component equals 76.5%.That is, if we compare conceptually similar rates we observe that the NELS-88 figure and the modified ECR differ by about one percentage point; none of the other measures described above so closely approximate the experience of the NELS-88 cohort. 9Specifically, if G10CTRL equals 1 and F4UNI2E is between 1 and 4. All analyses of the NELS-88 data are performed after weighting the data by F4PNLWT. 10 Students are counted as completers if they obtained some diploma in 1992, such that F4UNI2E equals 1 and YRRECY equals 1992. 11In-migrants who came to the U.S. after 1988 were not eligible to be counted among NELS-88 high school completers  1 the ECR is a conceptually unbiased estimate of the percentage of incoming public school 9 th graders in a particular state and in a particular year who complete high school.Figure 1  The fact that the ECR is first higher and then lower than the BCR over time in the U.S. reflects a trend toward high net in-migration in the United States.For any particular state in any particular year, whether the ECR yields substantially higher or lower estimates than the BCR (or other measures) is a largely a function of how much 9 th grade retention and net migration those states experience.For states with low 9 th grade retention rates and low net migration the ECR is virtually equivalent to the BCR (and to other measures).However, in states with high rates of 9 th grade retention and/or high levels of net migration the ECR can produce very different estimates.For example, Figure 2 plots the BCR and the ECR for Nevada for the graduating classes of 1973 through 2000.
Because Nevada has experienced very high rates of net in-migration annually-the population of 17 year olds is often more than 8% larger than the population of 14 year olds three years earlier-the ECR is usually five percentage points higher than the BCR (and occasionally as many as nine percentage points higher).In contrast, New York experienced moderate net out-migration until about 1980 and has experienced moderate net in-migration ever since then.The consequence, as shown in Figure 3, is that the ECR was sometimes more than five percentage points higher than the BCR in the late 1970s, whereas the two measures differ relatively little thereafter (because net in-migration is approximately off-set by 9 th grade retention after the early 1980s).
The point that the ECR can sometimes portray a very different picture about individual states' high school completion rates is made more dramatically by comparing states' relative rankings on the BCR and the ECR.The X-axis of Figure 4 arrays states according to their ranking on the ECR for the graduating class of 2000, where 1 represents the highest completion rate in 2000 (in New Jersey) and 51 represents the lowest completion rate (in South Carolina).The states' postal abbreviations are arrayed on the Y-axis according to the difference in relative rankings between the ECR and the BCR.For example, whereas Rhode Island ranked 32 nd on the BCR in 2000, it ranked 15 th on the ECR in that year-a difference of 17.How are states like Rhode Island, New Hampshire, Idaho, Connecticut, and Virginia doing relative to other states with respect to high school completion rates?The answer depends a great deal on one's choice of measure.
Figure 5 depicts the ECR for each state for the graduating class of 2000.South Carolina, Georgia, Arizona, Louisiana, and Florida had the lowest public high school completion rates in 2000-all below 57%-while New Jersey, Minnesota, Connecticut, North Dakota, and Nebraska had the highest rates-all above 81%.Figure 1 above showed a modest but steady decline in the ECR over time in the U.S. as a whole, and this trend holds in most individual states as well.Figure 6 demonstrates that high school completion rates declined in 46 states between 1973 and 2000, but that the size of the decline varied tremendously across states.Most states saw a decline in high school completion rates of between 5 and 15 percentage points, but South Dakota saw declines of more than 20 percentage points while Arkansas, the District of Columbia, New Jersey, Mississippi, and Virginia saw positive trends.

THE ECR AND PRIVATE SCHOOL ENROLLMENTS AND COMPLETIONS
The ECR represents the percentage of incoming public school 9 th graders in a particular state and in a particular year who complete public high school.The exclusion of enrolled private school students and graduates from the ECR could be problematic if there have been substantial changes over time in private high school enrollments and/or completions.This is particularly true if changes in private school enrollments and/or completions have occurred unevenly across socioeconomic and/or demographic groups or across geographic areas.For example, if racial inequalities in private school attendance and/or enrollment have widened over time, then the apparent decline in the ECR (and other public high school completion rates) over time may not be a reflection of real change in students' chances of completing public school.
To assess the extent to which changes in private school enrollments and completions are driving trends in the ECR, Figure 7  About 8% of high school students are enrolled in private schools.This figure has not changed perceptibly since at least 1977.Whites, students whose household head attended at least some college, and students in the New England and Middle Atlantic states are more likely than their peers to attend private high schools; none of these disparities in rates of private school attendance have changed perceptibly since at least 1977.Finally, as depicted in Panel D, there are notable regional differences in the rate at which high school completers graduate from private schools.However, neither the overall percentage of completers graduating from private schools nor regional differences in that percentage have changed since at least 1980.There are likely many factors behind changes over time and differences across states in high school completion rates, but changes in private school enrollments and completions likely play a small role.

THE ECR AND INTERNATIONAL IN-MIGRATION
The migration adjustment to the denominator in Equation 7 conceptually represents the net change in the size of a given cohort between ages 14 and 17; such changes can only be the result of migration and mortality.We begin with n 14 year olds in a particular state in a particular year.Over the next three years, some of the n die, some of the n leave the state, and individuals not counted among the original n move from outside of the state-either from other states or from abroad.A potential with this approach to adjusting for migration concerns young people who move to the U.S. from abroad between the ages of 14 and 17 but who do not enroll in public school.These students inflate the denominator of Equation 7, and thus reduce the ECR.However, if these young people never enter the public education system in the U.S. then the ECR may unfairly understate the public high school completion rate, especially in states that experience high levels of immigration.The size of this problem is an empirical question that is addressed in Table 3.
Columns 1 through 4 of Table 3 are based on data for 14 to 17 year olds from the 2000 U.S. Census 5% PUMS file.Column 1 reports the total number of 14 to 17 year olds in each state as of the 2000 enumeration.Column 2 reports the number of 14 to 17 year olds who were born outside of the U.S.-about 8.4% of all 14 to 17 year olds nationwide-and Column 3 reports the number of 14 to 17 year olds who were born outside of the U.S. and who came to the U.S. after age 13.About 17.8% of foreign born 14 to 17 year olds came to the U.S. after age 13.However, Column 4 shows that the vast majority of these young recent immigrants-about 76.9%-were enrolled in school in 2000.Nonetheless, in 2000 there were more than 75,000 people between the ages of 14 and 17 who immigrated after age 13 and who were not enrolled in school.If we assume that none of these young immigrants were ever enrolled in U.S. public schools, and remove them from the migration adjustment to the denominator in Equation 7, the ECR changes from 66.6% nationwide (Column 5) to 67.9% nationwide (Column 6)-an increase of 1.3 percentage points.The ECR understates the public school completion rate by less than 1 percentage point for most states, but by more than 2 percentage points in 8 states-all of which experience high levels of international immigration.The figures in Table 3 can only be reliably computed for 2000, and should serve as a cautionary note: The ECR modestly understates high school completion rates in states with many international immigrants who come to the U.S. between ages 14 and 17 and who do not enroll in school.

DOES THE CHOICE OF MEASURE DRIVE SUBSTANTIVE RESULTS?
As demonstrated above, conclusions about states' absolute and relative high school completion rates differ depending on how states' high school completion rates are measured.Beyond these descriptive differences, it is worth considering whether different state-level measures of high school completion perform differently in typical empirical analyses.To address this issue I have estimated models of the effect of (1) state-level poverty rates and (2) states' high school exit examination policies on state-level high school completion rates using alternate measures of the dependent variable.
Our data on annual state-level poverty rates are derived from U.S. Census Bureau data, and our information about whether states required students in particular graduating classes to pass an exit examination as a prerequisite for obtaining a high school diploma is derived from recent archival work by Warren and colleagues at the University of Minnesota (e.g., Warren, Jenkins, and Kulick 2004).
Briefly, we estimate a series of state and year fixed-effects models in which the 1,428 state-years be-tween 1973 and 2000 are our units of analyses.Our models include state and year fixed effects plus one time-varying covariate: either state-level poverty rates or states' high school exit examination policy.These analyses are by no means complete; they are simply designed to investigate whether substantive conclusions depend on how states' high school completion rates are operationalized.
Table 4 reports the results of these models.The models in each column use a different measure of state-level high school completion rates: a CPS status dropout rate, the BCR, the ACR, the CPI, and the ECR.Model A includes state-level poverty rates as the only time-varying covariate, and Model B includes states' high school exit examination policies as the only time-varying covariate.The results of Model A show that state poverty rates are related to high school dropout or completion rates-except when the ECR is used to measure states' high school completion rates.Although the BCR, ACR, CPI, and ECR are in the same metric (ranging from 0 to 100), the point estimates differ considerably depending on how the dependent variable is measured.The results of Model B show that high school exit examinations are associated with higher dropout rates and lower high school completion rates-except when the ACR is the measure of high school completion rates.
Again, the magnitude of the estimated effect of high school exit examinations also varies across outcome measures.In general, the results in Table 4 suggest that substantive results may depend in important ways on how state-level high school completion rates are measured.This highlights the importance of utilizing a measure that is conceptually sound and empirically unbiased.

DISCUSSION
In this paper I reviewed and critiqued existing state-level measures of high school completion that use CPS or CCD data.Measures based on the CPS are conceptually inappropriate for present purposes and are typically statistically unreliable because of small sample sizes in many states.
Measures based on Common Core Data (CCD) dropout information are unavailable for many states and have their own conceptual weakness.As shown in a series of simulations, existing measures based on CCD enrollment and completion data are systematically biased by migration, by changes in cohort size, and (except for the CPI) by grade retention.The BCR, ACR, and CPI misrepresent absolute rates of high school completion, states' relative standing with respect to high school completion rates, and trends over time in rates of high school completion.
After critiquing existing CCD-based measures I went on to describe a new measure-labeled an Estimated Completion Rate (ECR)-that uses these data to produce state-level high school completion rates for 1973 through 2000.The ECR conceptually represents the percentage of incoming public school 9 th graders in a particular state and in a particular year who complete high school.This measure is not influenced by changes over time in incoming cohort sizes, uses information from the U.S. Census Bureau to estimate states' net migration rates, and uses information from October CPS data to account for grade retention.While the ECR conceptually overcomes the systematic biases in other CCD-based high school completion rates that are produced by changes in cohort size, migration, and 9 th grade retention, its empirical accuracy hinges on the validity of the migration and 9 th grade retention adjustments (and, of course, on the quality of the CCD data themselves).However, as described above the ECR does a good job of approximating high school completion rates observed in longitudinal studies like the National Educational Longitudinal Study of 1988.There is certainly some degree of random error in the ECR estimates, but systematic biases in the ECR are far less than the systematic biases in alternate measures.Because different measures paint very different pictures of states' absolute and relative high school completion rates, and because (as shown in Table 4) the choice of measure of states' high school completion rates can drive substantive empirical results in important ways, it is important for researchers to utilize a measure of state-level high school completion rates that is conceptually sound and empirically unbiased.I argue that the ECR is the best choice in this regard.
While the ECR does a better job of accounting for sources of systematic bias that plague other measures that use the CCD, the ECR is certainly limited in a number of respects and will not be useful for all purposes.Because the ECR is a measure of the overall high school completion rate (not of the four-year completion rate) and because I do not restrict the numerator to regular diploma recipients, the ECR is not in line with the guidelines for measuring AYP in No Child Left Behind.
What is more, in this paper I have not computed the ECR separately by race/ethnicity (or even gender) because the CCD data do not contain race/ethnic-group specific completion counts for some states and because of the difficulties involved in producing valid and reliable group-specific migration and 9 th grade retention adjustments.For similar reasons I have not computed the ECR at geographic levels below the state, despite the need for local-level measures presented by the annual yearly progress requirements of the 2002 No Child Left Behind legislation.As described above, the ECR modestly understates high school completion rates in states with high levels of international inmigration.Finally, the ECR categorically treats GED recipients as individuals who have not completed high school.For many purposes this is a virtue of the ECR, but for other purposes it may be seen as a weakness.It is conceivable that the ECR could be amended to include GED recipients in the numerator using data from the GED Testing Service, 12 although it would be difficult to know which year GED recipients should be counted in the numerator of that revised ECR.
The ECR-like other CCD-based measures of high school completion-shows a disquieting trend: Since at least the early 1970s the rate at which incoming 9 th graders have gone on to complete high school in a timely fashion has declined steadily.In 8 states the high school completion rate declined by more than 15%; it increased in only four states and the District of Columbia.In the year 2000, only about two of every three public school students who might have completed high school actually did so.Any number of factors may account for this trend, including (but not limited to) 12 CCD data on numbers of GED recipients varies in quality from state to state and over time.changes in the demographic composition of students, increases in GED certification rates, and/or changes in a wide variety of education policies.In any case, careful investigation of this trend requires a conceptually sound and empirically valid measure of high school completion.Note: The ECR equals the number of high school completers (not including GED recipients) in spring of academic year X divided by the number of 9 th graders in fall of academic year X-3, with adjustments to the denominator to account for net migration and 9 th grade retention.See text for details.Note: The ECR equals the number of high school completers (not including GED recipients) in spring of academic year X divided by the number of 9 th graders in fall of academic year X-3, with adjustments to the denominator to account for net migration and 9 th grade retention.See text for details.

The
Common Core of Data, compiled by the National Center for Education Statistics (NCES), is the federal government's primary database on public elementary and secondary education.Each year the CCD survey collects information about all public elementary and secondary schools from local and state education agencies.One component of the CCD-the State Nonfiscal Survey-provides basic, annual information on public elementary and secondary school students and staff for each state and the District of Columbia.CCD data from the State Nonfiscal Survey includes counts of the number of students enrolled in each grade in the fall of each academic year and the number of students who earned regular diplomas, who earned other diplomas, and who completed high school in some other manner in the spring of each academic year.Although the State Nonfiscal Survey has collected counts of public school dropouts since the 1991-1992 academic year, as described below many states have not provided this information or have provided it in a manner inconsistent with the standard CCD definition of dropout (U.S.Department of Education 2000).

6
Throughout, I employ CPS-provided sampling weights that account for the probability of selection in to the CPS sample and that adjust for non-response U.S. Bureau of the Census.2002a.Current Population Survey, October 2000: School Enrollment Supplement File.Technical Documentation CPS-01.Washington, D.C.: U.S. Bureau of the Census..
California in 1969 is equal to an average of that estimated rate for 1969 through 1973.Although I use 1968 through 2000 October CPS data, 9 th grade retention rates can only be calculated beginning in 1969 (because I do not observe the age distribution of 8 th graders in 1967).Because academic year 1999-2000 is the last year in which I observe numbers of high school completers, I only need 9 th grade retention rates through 1996 (although because I use five year moving averages the rates for 1996 require data from 1996 through 2000). 7There are 18 state groups: (1) Connecticut, Massachusetts, Maine, New Hampshire, Vermont, and Rhode Island; (2) New York; (3) New Jersey; (4) Pennsylvania; (5) Ohio; (6) Indiana and Illinois; (7) Michigan and Wisconsin; (8) Minnesota, Iowa, Missouri, North Dakota, South Dakota, Kansas, and Nebraska; (9) The District of Columbia, Delaware, Maryland, Virginia, and West Virginia; (10) South Carolina, North Carolina, and Georgia; (11) Florida; (12) Kentucky and Tennessee; (13) Alabama and Mississippi; (14) Texas; (15) Arkansas, Lousiana, and Oklahoma; (16) Montana, Idaho, Wyoming, Colorado, New Mexico, Arizona, Utah, and Nevada; (17) California; (18) Washington, Oregon, Alaska, and Hawaii.
14 year olds in that state on July 1 three years earlier.These estimates are derived from published, annual state-by-age population estimates produced by the Population Division of the U.S. Bureau of the Census (U.S. Bureau of the Census 2001a; U.S. Bureau of the Census 2001b; U.S. Bureau of the Census 2002b) which are readily available for all years between 1970 and 2000.For example, there were 385,531 people age 14 in California in 1970.In that state in 1973 there were 389,109 people age 17-a 0.9% net increase.To improve the reliability of these estimates, I computed four year moving averages.The net migration estimate for California in 1980 thus represents the point estimates for 1980 through 1983. 8Again, these migration estimates are subject to random error; however, as I describe below they are largely unbiased.In any case, these estimates are certainly preferable to either ignoring migration or to using demonstrably biased estimates of migration.Because I am using population estimates from 1970 through 2000, I am unable to compute migration estimates for 1969 through 1972.For these years, I have imputed values based on models (estimated separately for each state) of trends in migration rates between 1973 and 1977.This technical issue aside, there are three potential problems with this technique for estimating migration rates.
depicts national high school completion rates as reflected by the BCR and by the ECR for the graduating classes of 1973 through 2000.Both estimators show that the high school completion rate in the United States declined in the late 1970s, remained stable during the 1980s and early 1990s, and declined again beginning in the early 1990s.Whereas more than three fourths of students completed high school in 1973, only two thirds did so in 2000.The ECR is 1.4 percentage points higher than the BCR in 1973, but 1.5 percentage points lower by 2000.While one or two percentage points may seem substantively trivial, keep in mind that more than three and half million students are in the denominator nationwide each year.One percentage point in these rates is a difference of about 35,000 young people nationwide.This means that in 2000 the BCR and ECR estimates of the number of non-completers differed by about 50,000 students nationwide.
depicts trends in the percentage of 9 th through 12 th graders who are enrolled in private schools by race (Panel A), household head's education (Panel B), and region (Panel C) and trends by geographic region in the percentage of high school completers who graduated from private schools (Panel D).Data for Panels A, B, and C are derived from October CPS data for 1977 through 2000; estimates are based on weighted data, and reflect three-year moving averages.Data for Panel D come from CCD counts of public school completers and counts of private school completers from various years of the Private School Universe Survey which is conducted periodically by the National Center for Education Statistics (U.S.Department of Education 2001b).

Figure 4 .
Figure 4. State Rankings on High School Completion Rate Measures, 2000

Figure 6 .
Figure 6.Changes in High School Graduation Rates (ECR) Between 1973 and 2000, by State

Table 2
reports the ECR by state and year of high school completion.As demonstrated in Table

Table 1 .
Various High School Completion Rates Under Different Assumptions: A Simulation C. Net Migration Rate of -1% Annually at Each Grade Level (No Change in Cohort Size, No Grade Retention, No Dropout)

Table 1 (
continued).Various High School Completion Rates Under Different Assumptions: A Simulation Table 2 (continued).High School Completion Rates (ECR), by State and Graduating Class Note : Data in Columns 1 through 4 are derived from the 2000 U.S. Census 5% PUMS file, and are weighted.

Table 3 .
Estimated Completion Rate in 2000, by State, Before and After Accounting for International In-Migrants Who Are Not Enrolled in School

Figure 7. Private School Enrollment Among 9 th to 12 th Graders, 1977 to 2000
. Percentage Enrolled in Private Schools, by Race B. Percentage Enrolled in Private Schools, by Parent's Education C. Percentage Enrolled in Private Schools, by Region D. Percentage of High School Graduates from Private Schools, by Region A